# Effects of Tooth Surface Crack Propagation on Meshing Stiffness and Vibration Characteristic of Spur Gear System

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Proposed Meshing Stiffness Calculation Model with Tooth Surface Crack

_{1}represents the crack length, L

_{2}stands for the crack width, and D refers to the crack depth.

_{b}, shear energy U

_{s}, and axial compressive energy U

_{a}deposited in a tooth can be expressed as [16,17],

_{b}, k

_{s}, and k

_{a}signify the bending stiffness, shear stiffness, and axial compressive stiffness, respectively. F refers to the meshing force acting on the gear tooth. E and G are the Young’s modulus and shear modulus, respectively, and it is assumed that the material of the gear is uniform and the Young’s modulus and shear modulus of its surface and internal parts are the same in this paper. d denotes the distance from the meshing point to the dedendum circle, and h is the distance from the meshing point to the center line of the gear tooth. F

_{a}and F

_{b}represent the horizontal and vertical components of the meshing force F, respectively, which are expressed as,

_{b}, Ks, and K

_{a}can be deduced as,

_{x}, A

_{x}, and I

_{x}are as follows,

_{x}is the cross-sectional height with x distance from the dedendum. R

_{b}and R

_{f}are the base circle radius and dedendum circle radius, respectively. α represents the gear rotation angle, α

_{2}and α

_{3}refer to half of tooth angle of the base circle and dedendum circle, respectively. A

_{x}is the area of the effective section and I

_{x}represents the area moment of inertia.

_{x}and I

_{x}remain unchanged when the contact line is located between G and S. Thus, the mesh stiffness of gear tooth pair in this area is the same as that of healthy gear teeth. As illustrated in Figure 3b, the crack part can still bear shear, axial compressive, and bending force when the contact line is located between S and P, and the mesh stiffness of gear teeth in this area is also the same as that of normal gear. When the contact line passes through the end point P of the crack, the crack turns to an open state due to the presence of meshing force, and the parameters A

_{x}and I

_{x}of the cracked tooth zone will decrease, which is presented in Figure 3c.

_{x}’ and the moment of inertia I

_{x}’ of the gear with tooth surface crack are calculated as follows,

_{x}

_{1}is the cross-sectional height with x

_{1}distance from the dedendum.

_{h}of healthy meshing tooth pairs is a constant throughout the meshing line. For the meshing gear teeth with surface crack, the crack will not affect the effective contact width of the gear teeth during the whole meshing process. Therefore, the Hertz contact stiffness of surface cracked teeth is consistent with the healthy case. Its expression is [34],

_{f}denotes the tooth fillet-foundation stiffness. The parameters μ

_{f}and S

_{f}are displayed in Figure 2 in [18], and the parameters L, Q, M, and P are functions of θ

_{f}and h

_{f}, which can be presented as [18],

_{f}, θ

_{f}, A

_{i}, B

_{i}, C

_{i}, D

_{i}, E

_{i}, and F

_{i}can be observed in [18].

## 3. Dynamic Modeling of Spur Gear System with Tooth Surface Crack

_{m}represents the mesh damping, e stands for the comprehensive meshing error, m

_{i}refers to the mass, J

_{i}is the mass moment of inertia, T

_{i}indicates the load torque, ω

_{i}represents the speed of rotation, and K

_{ix}and K

_{iy}are the radial stiffness of the bearing in the x and y direction, respectively. C

_{px}and C

_{py}signify the radial damping of the bearing in the x and y direction, respectively. The subscript i = p, g refers to the pinion and gear, respectively.

_{p}and θ

_{g}denote the angular displacement of the pinion and gear, respectively. x

_{p}and x

_{g}represent the lateral displacement of the pinion and gear along the x direction, respectively. y

_{p}and y

_{g}refer to the lateral displacement of the pinion and gear along the y direction, respectively.

## 4. Results and Discussions

#### 4.1. Effects of Single Tooth Surface Crack Parameter on Mesh Stiffness and Vibration Characteristics

#### 4.1.1. Effects of a Single Crack Parameter on Mesh Stiffness

_{1}= 1, 2, 3 mm, L

_{2}= 40 mm, D = 3 mm, and x

_{1}= 2 mm, the influence of the crack length (L

_{1}) on the meshing stiffness of single-tooth pair and double-tooth pairs is studied. It can be obtained that the total meshing stiffness of the cracked gear is smaller than that of the healthy gear at the beginning of the meshing process. The stiffness drop zone is enlarged with the increase in the crack length, while the decrease in the magnitude of the meshing stiffness under different crack lengths remains the same. The maximum ratios of the stiffness reduction of single and double tooth pairs are 8.2% and 7.0%, respectively.

_{2}) on the meshing stiffness of single-tooth pair and double-tooth pair is investigated when the surface crack parameters are set as L

_{1}= 2 mm, L

_{2}= 15, 30, 45 mm, D = 3 mm, and x

_{1}= 2 mm. It can be observed from the calculation results that the reduction in the magnitude of the meshing stiffness gradually goes up with the increase in the crack width, but the mesh period where the meshing stiffness decreases remain unchanged. The maximum ratios of the stiffness reduction of single and double tooth pairs are 13.0% and 8.7%, respectively.

_{1}= 2 mm, L

_{2}= 40 mm, D = 1, 2, 3 mm and x

_{1}= 2 mm. We can find in Figure 7 that the reduction in the magnitude of the meshing stiffness is gradually enlarged when the crack depth increases, while the areas where the mesh stiffness decreases remain the same. The maximum ratios of the stiffness reduction of single and double tooth pairs are 10.8% and 7.2%, respectively.

#### 4.1.2. Effects of a Single Crack Parameter on DTE

_{p}and T

_{p}are set to a fixed value equal to 900 rpm and 100 Nm, respectively. As shown in Figure 8a, the occurrence of cracks will increase the amplitude of DTE responses. The greater the crack length, the earlier the DTE value returns to the healthy tooth level. It can be discovered from Figure 8b that in the area affected by the surface crack failure, the value of DTE goes up with the increase in the crack width, and the DTE value is highest when L

_{2}= 45 mm. It can also be seen in Figure 8c that the value of DTE goes up with the crack propagation in the direction of crack depth, and the DTE value reaches the maximum when D = 3 mm.

#### 4.1.3. Effects of a Single Crack Parameter on Acceleration Response

#### 4.2. Effects of Surface Crack Propagation on Meshing Stiffness and Vibration Characteristics

_{i}(i = 1, 2, 3) denotes the crack vertex position in the process of crack propagation. In this study, the spalling failure is assumed to occur when the surface crack propagates to the position S

_{3}. Specific parameters of the surface crack propagation case are displayed in Table 2.

#### 4.2.1. Effects of Surface Crack Propagation Progress on Meshing Stiffness

#### 4.2.2. Effects of the Surface Crack Propagation Progress on DTE

_{m}and its harmonic frequencies (2f

_{m}, 3f

_{m}, …) in the spectrum diagrams of four fault cases, and the amplitudes of side frequency go up gradually with the surface crack propagation. In addition, the interval of the two adjacent side frequency components equals to the rotation frequency f

_{n}of the driving gear. It can be concluded that the surface crack propagation has a small influence on the time domain of DTE, but an obvious influence on the sideband components of the DTE spectrum, which is an important feature for the diagnosis of early spalling failure.

#### 4.2.3. Effects of the Surface Crack Propagation Progress on Acceleration Response

_{m}. k denotes the number of side frequency.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Chen, Z.G.; Zhai, W.M.; Wang, K.Y. Vibration feature evolution of locomotive with tooth root crack propagation of gear transmission system. Mech. Syst. Signal Process.
**2019**, 115, 29–44. [Google Scholar] [CrossRef] - Rashid HS, J.; Place, C.S.; Mba, D.; Lim, R.; Healey, A.; Kleine-Beek, W.; Romano, M. Helicopter MGB Oil System Failure Analysis Using Influence Diagrams and Random Failure Probabilities. Eng. Fail. Anal.
**2015**, 50, 7–19. [Google Scholar] [CrossRef] - Chen, K.K.; Huangfu, Y.F.; Ma, H.; Xu, Z.; Wen, B. Calculation of mesh stiffness of spur gears considering complex foundation types and crack propagation paths. Mech. Syst. Signal Process.
**2019**, 130, 273–292. [Google Scholar] [CrossRef] - Wang, L.M.; Shao, Y.M. Fault feature extraction of rotating machinery using a reweighted complete ensemble empirical mode decomposition with adaptive noise and demodulation analysis. Mech. Syst. Signal Process.
**2020**, 138, 106545. [Google Scholar] [CrossRef] - Chen, K.; Ma, H.; Che, L.Y.; Li, Z.W.; Wen, B. Comparison of meshing characteristics of helical gears with spalling fault using analytical and finite-element methods. Mech. Syst. Signal Process.
**2019**, 121, 279–298. [Google Scholar] [CrossRef] - Maláková, S.; Pukár, M.; Frankovsk, P.; Sivák, S.; Palko, M. Meshing Stiffness—A Parameter Affecting the Emission of Gearboxes. Appl. Sci. Basel
**2020**, 10, 8678. [Google Scholar] [CrossRef] - Mohammed, O.D.; Rantatalo, M. Gear fault models and dynamics-based modelling for gear fault detection—A review. Eng. Fail. Anal.
**2020**, 117, 104798. [Google Scholar] [CrossRef] - Elyousfi, B.; Soualhi, A.; Medjaher, K.; Guillet, F. New approach for gear mesh stiffness evaluation of spur gears with surface defects. Eng. Fail. Anal.
**2020**, 116, 104740. [Google Scholar] [CrossRef] - Shi, L.J.; Wen, J.; Pan, B.S.; Xiang, Y.; Lin, C. Dynamic characteristics of a gear system with double-teeth spalling fault and its fault feature analysis. Appl. Sci. Basel
**2020**, 10, 7058. [Google Scholar] [CrossRef] - Huangfu, Y.F.; Chen, K.K.; Ma, H.; Li, X.; Han, H.; Zhao, Z. Meshing and dynamic characteristics analysis of spalled gear systems: A theoretical and experimental study. Mech. Syst. Signal Process.
**2020**, 139, 106640. [Google Scholar] [CrossRef] - Du, W.T.; Zeng, Q.; Shao, Y.M.; Wang, L.M.; Ding, X.X. Multi-Scale Demodulation for Fault Diagnosis Based on a Weighted-EMD De-Noising Technique and Time-Frequency Envelope Analysis. Appl. Sci. Basel
**2020**, 10, 7796. [Google Scholar] [CrossRef] - Chaari, F.; Fakhfakh, T.; Haddar, M. Analytical modelling of spur gear tooth crack and influence on gear mesh stiffness. Eur. J. Mech. A Solids
**2009**, 28, 461–468. [Google Scholar] [CrossRef] - Weber, C. The Deformation of Loaded Gears and the Effect on Their Load Carrying Capacity, Sponsored Research (Germany). Br. Dep. Sci. Ind. Res.
**1949**, 3, 22–30. [Google Scholar] - Cornell, R.W. Compliance and stress sensitivity of spur gear teeth. J. Mech. Transm. Autom.
**1981**, 103, 447–459. [Google Scholar] [CrossRef] [Green Version] - Kasuba, R.; Evans, J.W. An extended model for determining dynamic loads in spur gearing. J. Mech. Des. Trans. Am. Soc. Mech. Eng.
**1981**, 103, 398–409. [Google Scholar] [CrossRef] - Yang, D.C.H.; Lin, J.Y. Hertzian damping, tooth friction and bending elasticity in gear impact dynamics. J. Mech. Transm. Autom.
**1987**, 109, 189–196. [Google Scholar] [CrossRef] - Tian, X.H. Dynamic Simulation for System Response of Gearbox Including Localized Gear Faults. Master’s Thesis, University of Alberta, Edmonton, AB, Canada, 2004. [Google Scholar]
- Sainsot, P.; Velex, P.; Duverger, O. Contribution of gear body to tooth deflections a new bidimensional analytical formula. J. Mech. Des. ASME
**2004**, 126, 748–752. [Google Scholar] [CrossRef] - Chen, Z.G.; Shao, Y.M. Mesh stiffness calculation of a spur gear pair with tooth profile modification and tooth root crack. Mech. Mach. Theory
**2013**, 62, 63–74. [Google Scholar] [CrossRef] - Lei, Y.G.; Liu, Z.Y.; Wang, D.L. A probability distribution model of tooth pits for evaluating time-varying mesh stiffness of pitting gears. Mech. Syst. Signal Process.
**2018**, 106, 355–366. [Google Scholar] [CrossRef] - Yu, W.N.; Mechefske, C.K.; Timusk, M. A new dynamic model of a cylindrical gear pair with localized spalling defects. Nonlinear Dyn.
**2018**, 91, 2077–2095. [Google Scholar] [CrossRef] - Chaari, F.; Baccar, W.; Abbes, M.S.; Haddar, M. Effect of spalling or tooth breakage on gearmesh stiffness and dynamic response of a one-stage spur gear transmission. Eur. J. Mech. A Solids
**2008**, 27, 691–705. [Google Scholar] [CrossRef] - Ma, R.; Chen, Y.S.; Cao, Q.J. Research on dynamics and fault mechanism of spur gear pair with spalling defect. J. Sound Vib.
**2012**, 331, 2097–2109. [Google Scholar] [CrossRef] - Ma, R.; Chen, Y.S. Research on the dynamic mechanism of the gear system with local crack and spalling failure. Eng. Fail. Anal.
**2012**, 26, 12–20. [Google Scholar] [CrossRef] - Han, L.; Qi, H. Influences of tooth spalling or local breakage on time-varying mesh stiffness of helical gears. Eng. Fail. Anal.
**2017**, 79, 75–88. [Google Scholar] [CrossRef] - Shao, Y.M.; Wang, X.L.; Liu, J.; Chen, Z.G. Time-varying stiffness model and dynamic response characteristics of gears with tooth surface spalling and edge contact. J. Vib. Shock
**2014**, 33, 8–14. (In Chinese) [Google Scholar] - Fakhfakh, T.; Chaari, F.; Haddar, M. Numerical and experimental analysis of a gear system with teeth defects. Int. J. Adv. Manuf. Technol.
**2005**, 25, 542–550. [Google Scholar] [CrossRef] - Ma, H.; Li, Z.W.; Feng, M.J.; Feng, R.; Wen, B. Time-varying mesh stiffness calculation of spur gears with spalling defect. Eng. Fail. Anal.
**2016**, 66, 166–176. [Google Scholar] [CrossRef] - Saxena, A.; Parey, A.; Chouksey, M. Time varying mesh stiffness calculation of spur gear pair considering sliding friction and spalling defects. Eng. Fail. Anal.
**2016**, 70, 200–211. [Google Scholar] [CrossRef] - Jiang, H.J.; Shao, Y.M.; Mechefske, C.K. Dynamic characteristics of helical gears under sliding friction with spalling defect. Eng. Fail. Anal.
**2014**, 39, 92–107. [Google Scholar] [CrossRef] - Luo, Y.; Baddour, N.; Liang, M. Dynamical modeling and experimental validation for tooth pitting and spalling in spur gears. Mech. Syst. Signal Process.
**2019**, 119, 155–181. [Google Scholar] [CrossRef] - Cheng, S.H.; Hu, H.T.; Shi, Q. Analysis on the Reason of Crack on Carburized Gear Tooth Surface. China Heavy Equip.
**2016**, 3, 41–45. (In Chinese) [Google Scholar] - Ding, Y.; Rieger, N.F. Spalling formation mechanism for gears. Wear
**2003**, 254, 1307–1317. [Google Scholar] [CrossRef] - Gu, X.; Velex, P.; Sainsot, P.; Bruyère, J. Analytical investigations on the mesh stiffness function of solid spur and helical gears. J. Mech. Des.
**2015**, 137, 063301. [Google Scholar] [CrossRef] - Ding, X.X.; He, Q.B.; Shao, Y.M.; Huang, W. Transient Feature Extraction Based on Time-Frequency Manifold Image Synthesis for Machinery Fault Diagnosis. IEEE Trans. Instrum. Meas.
**2019**, 68, 1–14. [Google Scholar] [CrossRef]

**Figure 3.**Meshing positions of gear tooth with surface crack fault: (

**a**) left of the fault zone, (

**b**) above the fault zone, (

**c**) right of the fault zone.

**Figure 5.**Effects of surface crack length on meshing stiffness: (

**a**) single-tooth pair, (

**b**) double-tooth pairs.

**Figure 6.**Effects of surface crack width on meshing stiffness: (

**a**) single-tooth pair, (

**b**) double-tooth pairs.

**Figure 7.**Effects of surface crack depth on meshing stiffness: (

**a**) single-tooth pair, (

**b**) double-tooth pairs.

**Figure 9.**Effects of crack parameters on the acceleration responses: (

**a**) length, (

**b**) width, (

**c**) depth.

**Figure 11.**Effects of surface crack propagation on meshing stiffness: (

**a**) single-tooth pair, (

**b**) double-tooth pairs.

**Figure 13.**DTE spectra of different fault cases: (

**a**) case #1, (

**b**) case #2, (

**c**) case #3, (

**d**) case #4.

**Figure 15.**Acceleration spectra of different fault cases: (

**a**) case #1, (

**b**) case #2, (

**c**) case #3, (

**d**) case #4.

Parameter | Pinion | Gear |
---|---|---|

Teeth number | 23 | 39 |

Module (mm) | 3 | 3 |

Teeth width (mm) | 50 | 50 |

Pressure angle (^{o}) | 20 | 20 |

Poisson’s ratio | 0.3 | 0.3 |

Addendum coefficient | 1 | 1 |

Dedendum coefficient | 0.25 | 0.25 |

Hub radius (mm) | 25 | 25 |

Young’s modulus E (MPa) | 2.06 × 10^{5} | 2.06 × 10^{5} |

Mass (kg) | 1.32 | 3.16 |

Mass moment of inertia (kg·m^{2}) | 9.8 × 10^{−4} | 68 × 10^{−4} |

Bearing radial stiffness (N/m) | K_{px} = K_{py} = 5.8 × 10^{8} | K_{gx} = K_{g y} = 5.8 × 10^{8} |

Bearing radial damping (N·s/m) | C_{px} = C_{py} = 5 × 10^{3} | C_{gx} = C_{gy} = 5 × 10^{3} |

Case | Failure Degree | Crack Position | Crack Parameters | |||
---|---|---|---|---|---|---|

L_{1} | L_{2} | D | x_{1} | |||

Case #0 | Healthy | P | 0 | 0 | 0 | 5 |

Case #1 | 33.3% crack | S_{1} | 1 | 30 | 0.73 | 4 |

Case #2 | 66.7% crack | S_{2} | 2 | 30 | 1.39 | 3 |

Case #3 | 100% crack | S_{3} | 3 | 30 | 2 | 2 |

Case #4 | Spalling | S_{3} | 3 | 30 | 2 | 2 |

Response Type | Fault Case | BAR (dB) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Harmonic Order | Average Value | ||||||||||||

f_{m} | 2f_{m} | 3f_{m} | 4f_{m} | 5f_{m} | 6f_{m} | 7f_{m} | 8f_{m} | 9f_{m} | 10f_{m} | 11f_{m} | |||

DTE | Case #1 | −4.4 | −4.3 | −3.3 | −4.5 | −4.1 | −3.0 | −4.1 | −3.8 | −3.1 | −4.3 | −3.5 | −3.9 |

Case #2 | −4.1 | −4.0 | −2.8 | −4.2 | −3.8 | −2.7 | −3.9 | −3.6 | −3.1 | −4.1 | −3.3 | −3.6 | |

Case #3 | −3.8 | −3.6 | −2.4 | −3.9 | −3.5 | −2.3 | −3.6 | −3.3 | −2.7 | −3.7 | −3.0 | −3.3 | |

Case #4 | −1.3 | −1.5 | −0.8 | −1.7 | −1.3 | −0.5 | −1.3 | −1.2 | −1.2 | −1.6 | −1.1 | −1.2 | |

Acceleration | Case #1 | 0.4 | −1.3 | −0.9 | −2.8 | −2.7 | −2.5 | −3.7 | −3.5 | −3.4 | −3.4 | −2.4 | −2.4 |

Case #2 | 0.4 | −1.3 | −0.9 | −2.7 | −2.5 | −2.3 | −3.6 | −3.4 | −3.3 | −3.3 | −2.4 | −2.3 | |

Case #3 | 0.5 | −1.2 | −0.8 | −2.6 | −2.4 | −2.2 | −3.3 | −3.1 | −3.0 | −3.0 | −2.4 | −2.1 | |

Case #4 | 0.7 | −1.0 | −0.1 | −1.0 | −1.0 | −0.6 | −1.2 | −1.3 | −1.6 | −1.4 | −0.9 | −0.9 |

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**MDPI and ACS Style**

Yang, L.-t.; Shao, Y.-m.; Jiang, W.-w.; Zhang, L.-k.; Wang, L.-m.; Xu, J.
Effects of Tooth Surface Crack Propagation on Meshing Stiffness and Vibration Characteristic of Spur Gear System. *Appl. Sci.* **2021**, *11*, 1968.
https://doi.org/10.3390/app11041968

**AMA Style**

Yang L-t, Shao Y-m, Jiang W-w, Zhang L-k, Wang L-m, Xu J.
Effects of Tooth Surface Crack Propagation on Meshing Stiffness and Vibration Characteristic of Spur Gear System. *Applied Sciences*. 2021; 11(4):1968.
https://doi.org/10.3390/app11041968

**Chicago/Turabian Style**

Yang, Lan-tao, Yi-min Shao, Wei-wei Jiang, Lu-ke Zhang, Li-ming Wang, and Jin Xu.
2021. "Effects of Tooth Surface Crack Propagation on Meshing Stiffness and Vibration Characteristic of Spur Gear System" *Applied Sciences* 11, no. 4: 1968.
https://doi.org/10.3390/app11041968